Statistical Dispersion - Measures of Statistical Dispersion

Measures of Statistical Dispersion

A measure of statistical dispersion is a nonnegative real number that is zero if all the data are the same and increases as the data become more diverse.

Most measures of dispersion have the same scale as the quantity being measured. In other words, if the measurements have units such as metres or seconds, the measure of dispersion has the same units. Such measures of dispersion include:

• Standard deviation
• Interquartile range or Interdecile range
• Range
• Mean difference
• Median absolute deviation
• Average absolute deviation (or simply called average deviation)
• Distance standard deviation

These are frequently used (together with scale factors) as estimators of scale parameters, in which capacity they are called estimates of scale. Robust measures of scale are those unaffected by a small number of outliers.

All the above measures of statistical dispersion have the useful property that they are location-invariant, as well as linear in scale. So if a random variable X has a dispersion of S_X then a linear transformation Y = aX + b for real a and b should have dispersion S_Y = |a|S_X.

Other measures of dispersion are dimensionless (scale-free). In other words, they have no units even if the variable itself has units. These include:

• Coefficient of variation
• Quartile coefficient of dispersion
• Relative mean difference, equal to twice the Gini coefficient

There are other measures of dispersion:

• Variance (the square of the standard deviation) — location-invariant but not linear in scale.
• Variance-to-mean ratio — mostly used for count data when the term coefficient of dispersion is used and when this ratio is dimensionless, as count data are themselves dimensionless: otherwise this is not scale-free.

Some measures of dispersion have specialized purposes, among them the Allan variance and the Hadamard variance.

For categorical variables, it is less common to measure dispersion by a single number. See qualitative variation. One measure that does so is the discrete entropy.